In WMF production it's expected that services propagate headers used for tracing microservice sub-requests back to the root user request.
Historically this meant only x-request-id being passed through the RPC call chain in this way.
However we've recently added also the W3C-standard headers traceparent and tracestate, which are used by our distributed tracing implementation.
From searching the codebase, CirrusSearch doesn't seem to propagate any of these headers.
In Mediawiki core we now have Wikimedia/Http/TelemetryHeadersInterface, which allows for pluggable implementations that extract the necessary headers, and currently only has one implementation: Mediawiki/Http/Telemetry https://gerrit.wikimedia.org/g/mediawiki/core/+/master/includes/http/Telemetry.php
I think it should be enough to call Telemetry::getInstance()::getRequestHeaders() and add the returned array to all HTTP requests sent while CirrusSearch handles an incoming request.